Math Art: Latin Square

According to Wikipedia: "A Latin square is an n × n table filled with n different symbols in such a way that each symbol occurs exactly once in each row and exactly once in each column." It is like a Sudoku board, but disregarding the nine 3 × 3 squares.

Latin Squares

We can also make Orthogonal Latin Squares (a.k.a. Greaco-Latin Squares) which are defined by Wikipedia as: "An orthogonal Latin squares of order n over two sets S and T, each consisting of n symbols, is an n × n arrangement of cells, each cell containing an ordered pair (s,t), where sS and a tT, such that

  • every row and every column contains exactly one sS and exactly one tT, and
  • no two cells contain the same ordered pair of symbols."

You can think of this as one Latin square on top of another in which each ordered pair at each position on the Latin square is unique. Here are a couple examples. The larger squares make a Latin square, the smaller squares make a Latin Square, and no two ordered pairs are the same.

3 × 3 Orthogonal Latin Squares
4 × 4 Orthogonal Latin Squares

To take it one step further, we can have Mutually Orthogonal Latin Squares (MOLS). This involves a set of Latin Squares in which each pair of Latin Squares is orthogonal. The following image contains 3 Latin squares (small, medium, and large squares) and each pair of Latin Squares is orthogonal.

4 × 4 Mutually Orthogonal Latin Squares

Here is an image that I made of eight 9 × 9 MOLS:

9 × 9 Mutually Orthogonal Latin Squares

Further Reading